Spahn HTT, 2. fibrations of simplicial sets (changes)

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This is a subentry of a reading guide to HTT.

Contents

2.0 (Overview and glossary of fibrations and anodyne morphisms)

In particular the following notions are important:

• Left fibrations: These are the higher analogs of fibrations in groupoids.

• Anodyne morphisms: These are morphisms possessing the left lifting property with respect to all Kan fibrations. Anodyne morphisms are precisely the acyclic cofibrations in the standard model structure on simplicial sets?.

• The covariant model structure: This is a model structure on the over category $sSet/S$. It is a model structure for left fibrations. It is functorial in $S$. It is called covariant since by the $\infty$-Grothendieck construction $Fib_l (S)$ is associated to the category of covariant functors $Fun(S,\infty Grpd)$. There is also a notion of contravariant model structure where right fibrations $Fib_r (S)$ is associated to $Fun(S^{op},\infty Grpd)$.

• The Joyal model structure: Precisely $\infty$-categories are fibrant with respect to this model structure.

• inner fibration and minimal fibration: These are used to develop a theory of n-categories.

• cartesian fibration: These are higher analogs to Grothendieck fibrations (not necessarily in groupoids). These are defined with respect to cartesian morphisms.

• categorical fibration?: These are the fibrations in the Joyal model structure on $sSet$ (also called model structure for quasi-categories): Morphisms of simplicial sets possessing the right lifting properties against acyclic cofibrations in this model structure. Here the cofibrations are the monomorphisms and the weak equivalences are called weak categorical equivalences. Categorical fibrations have no intrinsic meaning in $\infty$-category theory. Fibrant objects in this model structure are precisely $\infty$-categories.

For a general overview see model structure on simplicial sets and fibration of quasi-categories.

Definition

A morphism of simplicial sets is called

1. a Kan fibration if it has the right lifting property with respect to every horn inclusion.

2. a trivial fibration if it has the right lifting property with respect to every boundary inclusion $\partial \Delta[n]\hookrightarrow \Delta[n]$.

3. a left fibration if it has the right lifting property with respect to every horn inclusion except the right outer one.

4. a left fibration if it has the right lifting property with respect to every horn inclusion except the left outer one.

5. a left fibration if it has the right lifting property with respect to every inner horn inclusion.

6. left anodyne if it has the left lifing property with respect to every left fibration.

7. right anodyne if it has the left lifing property with respect to every right fibration.

8. inner anodyne if it has the left lifing property with respect to every inner fibration.

9. minimal fibration roughly said, when the morphism is determined by its values on the boundaries.

10. cartesian fibration

11. cocartesian fibration

12. categorical fibration

We have the following relations of kinds of fibrations where the arrows indicate implication (e.g. an acyclic fibration is a Kan fibration).

$\array{ &acyclic\; fibration \\ &\downarrow \\ &Kan\; fibration \\ \swarrow&&\searrow \\ left \;fibration&&right \;fibration \\ \downarrow&&\downarrow \\ cocartesian\; fibration&&\cartesian\; fibration \\ \searrow&&\swarrow \\ &categorical\; fibration \\ &\downarrow \\ &inner \;fibration }$
Remark

We have the following intuition in regard to these types of fibrations

1. Right fibrations are the $\infty$-categorical analog of fibrations in groupoids.

2. Left fibrations are the $\infty$-categorical analog of cofibrations in groupoids.

3. Cartesian fibrations are the $\infty$-categorical analog of fibrations (not necessarily in groupoids).

Remark

(anodyne morphisms)

1. Every map of simplicial sets admits a factorization into an anodyne ((left anodyne, right anodyne, inner anodyne, a coﬁbration) followed by a Kan ﬁbration (left ﬁbration, right ﬁbration, inner ﬁbration, trivial ﬁbration).

2. The theory of left fibrations and left anodyne morphisms is dual to that of right fibrations and right anodyne morphisms; i.e. if $S\to T$ is a left fibration (left anodyne morphisms) iff the induced map $S^{op}\to T^{op}$ right fibrations and right anodyne morphisms.

2.1 Left fibrations

2.1.1 Left fibrations in classical category theory

Left fibrations are analogs to fibrations in groupoids.

Proposition 2.1.1.3 +

Let $F:C\to D$ be a functor between categories. The $F$ is a fibrations in groupoids iff the induced map $N(F):N(C)\to N(D)$ is a left fibration of simplicial sets.

Compare the following lemma with Proposition 2.1.3.1

Lemma 2.1.1.4 +

Let $q:X\to S$ be a left fibration of simplicial sets. The assignment

$\begin{cases} hS\to H \\ s\mapsto X_s \\f\mapsto f_1 \end{cases}$

2.1.2 Stability properties of left fibrations

The aim of this section is to show that left fibrations exist in abundance.

The following remark follows from a theorem of Joyal (not displayed here).

Remark (2.1.2.2, 2.1.2.9)

(left fibrations) 1. The projection from the over category is a left fibration.

1.The property of being a left fibration is stable under forming functor categories.

2.1.3 A characterization of Kan fibrations

Compare the following proposition with Lemma 2.1.1.4.

Proposition 2.1.3.1 +

Let $p:S\to T$ be a left fibration of simplicial sets. Then the following statements are equivalent

1. $p$ is a Kan fibration.

2. For every edge $f:t\to t^\prime$ in $T$, the map $f_!:S_t\to S_{t^\prime}$ is an isomorphism in the homotopy category of spaces.

2.1.4 The covariant model structure

This section is a preparation for the Grothendieck consruction ( more precisely for the $(\infty,0)$-Grothendieck construction) for $\infty$-categories. See also Grothendieck construction in HTT.

Requisites are the discussion of model structures on simplicial sets HTT, A.2 and that of simplicially enriched categories HTT, A.3.

The covariant model structure is a ‘’relative model structure’‘ in that it is a model structure on an overcategory. In HTT the theory of model structures on over categories is developed only for the case $sSet/S$ of simplicial sets. This model is ‘’functorial in $S$’‘ in the sense that every morphism $j:S\to S^\prime$ of simplicial sets induces a Quillen adjunction $(j_!\dashv j^*):sSet/S^\prime\stackrel{j^*}{\to}sSet/S$; see Proposition 2.1.4.10.

For the following definition recall the definition of the right cone $X^\triangleright:=X\star \Delta^0$ and the left cone $X^\triangleleft:=\Delta^0\star X$ of a simplicial set, where $\star$ denotes the join of simplicial sets.

Definition 2.1.4.2

Let $f:X\to S$ be a map of simplicial sets.

The simplicial set $C^\triangleleft(f):=S\coprod_X X^\triangleleft$ is called the left cone of $f$.

Remark 2.1.4.3

Let $f:X\to Y$ be a map of simplicial sets.

Then there is a canonical monomorphism $S\to C^\triangleleft(f)$.

We hence regard $S\subseteq C^\triangleleft(f)$ as a simplicial subset. There is precisely one vertex of $C^\triangleleft(f)$ which does not belong to $S$. We call this point the cone point of $C^\triangleleft(f)$.

Definition 2.1.4.5

(the covariant model structure aka. model structure for left fibrations)

Let $S$ be a simplicial set . A morphism $f:X\to Y$ in $sSet / S$ is called a

(C) covariant cofibration if it is a monomorphism of simplicial sets.

(W) a covariant weak equivalence if the induced map

$X^\triangleleft\coprod_X S\to Y^\triangleleft\coprod_Y S$

is a categorical weak equivalence.

(F) covariant fibration if it has the right lifting property with respect to every map wich is both a covariant cofibration and a covariant equivalence.

Lemma 2.1.4.6

Every left anodyne map is a covariant equivalence.

Proposition 2.1.4.7

The covariant model structure determines a left proper, combinatorial model structure on $sSet_{/S}$

Proposition 2.1.4.9

Every covariant fibration is a left fibration of simplicial sets

Proposition 2.1.4.10

(the covariant model structure is functorial in $S$)

For every map $j:S\to S^\prime$ of simplicial sets we have a Quillen adjunction

$(j_!\dashv j^*):sSet/S^\prime\stackrel{j^*}{\to}sSet/S$

with respect to the covariant model structure where $j_!:=f\circ(-):sSet/S\to sSet/S^\prime$ is the postcomposition-with-$f$ functor and its right adjoint is given by $j^* X^\prime=X^\prime\times_{S^\prime}S$.

Remark 2.1.4.12

There is also a contravariant model structure

2.2 Simplicial categories and $\infty$-categories

2.2.5 The Joyal model structure

Requisite: Theorem 2.4.6.1: Let Joyal model structure in HTT$X$ be a simplicial set. Then $X$ is fibrant in the Joyal model structure iff $X$ is an $\infty$-category.

Theorem 2.2.5.1

The exists a left proper, combinatorial model structure on the category of simplicial sets such that

(C) Cofibrations are precisely monomorphisms

(W) A map $p$ is a categorical equivalence iff $S(p)$ is an equivalence of simplicial categories. Where $S:sSet\to sCat$ denotes the functor induced via Kan extension by the cosimplicial object $\mathfrak{C}:\Delta\to sCat$, Definition 1.1.5.1, HTT.

(transclusion:

Proposition A.3.2.4

Let $S$ be a combinatorial monoidal model category. Let every object of $S$ be cofibrant. Let the collection of all weak equivalences in $S$ be stable under filtered colimits.

Then there exists a left proper, combinatorial model structure on $S Cat$ such that:

(C) The class of cofibrations in $S Cat$ is the smallest weakly saturated class of morphisms containing the set of morphisms $C_0$ defined in A.3.2.3. ($C_0$ is some class of ‘’indicating morphisms’’).

(W) The weak equivalences in $S Cat$ are those functors $F:C\to D$ which are essentially surjective on the level of homotopy categories and such that $Map_C(X,Y)\to Map_{C^\prime}(F(X),F(Y))$ for every $X,Y\in C$.

Recall that $sSet$ equipped with the can model structure is an excellent model category.

Theorem 3.2.24

Let $S$ be an excellent model category. Then:

1. An $S$-enriched category $C$ is a fibrant object of $sSet Cat$ iff it is locally fibrant: i.e. for all $X,Y\in C$ the hom object $Map_C (X,Y)\in S$ is fibrant.

2. Let $F:C\to D$ be a $S$-enriched functor where $D$ is a fibrant object of $sSet Cat$. Then $F$ is a fibration iff $F$ is a local fibration.

) here: give proof of Proposition 1.2.7.3

2.3 Inner fibrations

Every functor $f:C\to D$ of classical categories induces an inner fibration $N(C)\to N(D)$

Corollary 2.3.2.2

$Fun(\Delta,C)\to Fun(\Lambda_1,C)$ is a trivial fibration.

Every $\infty$-category is categorial equivalenct to a minimal $\infty$-category.

2.3.3. Minimal inner fibrations

Definition

Let

$\array{ A&\stackrel{u}{\to}&X \\ \downarrow^i&&\downarrow^p \\ B&\stackrel{v}{\to}&S}$

denote a lifting problem. Then putative solutions $f,g$ of this lifting problem are called homotopic relative $A$ over $S$ if they are equivalent as objects in the fiber of the map

$X^B\to X^A\times_{S^A}S^B$

Equivalently $f,g$ are homotopic relative $A$ over $B$ if there is a map

$F:B\times \Delta\to X$

such that

$F|B\times\{0\}=f$

$F|B\times\{1\}=g$

$p\circ F=v\circ \pi_B$

$F\circ(i\times id_{\Delta})=u\circ\pi_A$

$F|\{b\}\times \Delta$

and $F|\{b\}\times\Delta$ is an equivalence in the $\infty$-category $X_{v(b)}$ for every vertex $b$ of $B$.

Definition 2.3.3.1

Let $p : X \to S$ be an inner ﬁbration of simplicial sets. $p$ is called minimal inner fibration if $f = f^\prime$ for every pair of maps $f , f ^\prime : \Delta[n] \to X$ which are homotopic relative to $\partial \Delta[n]$ over $S$ .

An $\infty$-category $C$ is called minimal $\infty$-category if $C\to *$ is minimal.

(…)

Every $\infty$-category is equivalent to a minimal $\infty$-category.

2.3.4 Theory of $n$-categories

Proposition 2.3.4.19

Proposition 2.3.4.5: For a simplicial set $X$ the following statements are equivalent:

1. the unit $u:X\to N(hX)$ is an isomorphism of simplicial sets.

2. There is small category $C$ and an isomorphism of simpliial sets $X\simeq N(C)$.

3. $S$ is a 1-category.

Proposition 2.3.4.7

Let $C$ be an $\infty$-category. Let $n\ge -1$. Then the following statements are equivalent:

1. $C$ is an $n$-category.

2. For every simplicial set $K$ and every pair of maps $f,g:K\to C$ such that $f| sk^n K$ and $g|sk^n K$ are homotopic relative to $sk^{n-1}K$, we have $f=g$.

Corollary 2.3.4.8

Let $C$ be an $n$-category and let $X$ be a simplicial set. Then $Fun(X,C)$ is an $n$-category.

Proposition 2.3.4.12

Let $C$ be an $\infty$-category. Let $n\ge 1$.

1. There exists a simplicial set $h_n C$ with the following universal mapping property: $Fun(K,h_n C)=[K,C]/\sim$.

2. $h_n C$ is an $n$-category.

3. If $C$ is an $n$-category, then the natural map $\Theta:C\to h_n C$ is an isomorphism.

4. For every $n$-category $D$, composition with $\Theta$ is an isomorphism of simplicial sets $Fun(h_n C,D)\to Fun(C,D)$.

Proposition 2.3.4.18

Let $C$ be an $\infty$-category and let $n\ge -1$. The the following statements are equivalent:

1. There exists a minimal model $C^\prime\subseteq C$ such that $C^\prime$ is an $n$-category.

2. There exists a categorical equivalence $D\to C$, where $D$ is an $n$-category.

3. For every pair of objects $X,Y\in C$, the mapping space $Map_C(X,Y)\in H$ is $(n-1)$-truncated.

Corollary 2.3.4.19

Let $X$ be a Kan complex. Then is is equivalent to an $n$-category iff it is $n$-truncated.

2.4 Cartesian fibrations

2.4.1 Cartesian morphisms

Definiton 2.4.1.1

Let $p:X\to S$ be an inner fibrations of simplicial sets. Let $f:x\to y$ be an edge in $X$. Then $f$ is called $p$-cartesian if the induced map

$X_{/f}\to X_{/y}\times_{S_{p(y)}} S_{/p(f)}$

is a trivial Kan fibration.

Proposition 2.4.1.3
1. Every edge of a simplicial set is $p$ cartesian for an isomorphism.

2. Let $p$ be an inner fibration, let $q$ be the pullback of $p$ (which s then also an inner fibration). Then an edge is $p$ cartesian if ‘’its pullback’‘ is $q$-cartesian.

3. (…)

Corollary 2.4.1.6

Let $p:C\to D$ be an inner fibration between $\infty$-categories. Every identity morphism of $C$ (i.e. every degenerate edge of $C$) is $p$-cartesian.

Proposition 2.4.1.7

(left cancellation) Let $p:C\to D$ be an inner fibration between simplicial sets. Let

$\array{ &C_1& \\ {}_f\nearrow&\searrow^g \\ C_0&\stackrel{h}{\to}&C_2 }$

Let $g$ be $p$-cartesian. Then $f$ is $p$-cartesian iff $h$ is $p$-cartesian.

Proposition 2.4.1.10

Let $F:C\to D$ be a functor between simplicial categories. Let $C$ and $D$ be fibrant. Let for every pair of object $c, c^\prime\in C$ be the associated map

$Map_C(c,c^\prime)\to Map_D(F(c),F(c^\prime))$

be a Kan fibration. Then:

1. The associated map $q:N(C)\to N(D)$ is an inner fibration between $\infty$-categories.

2. A morphism $f:c^\prime\to d$ in $C$ is $q$-cartesian iff for every $e\in C$ the diagram of simplicial sets

$\array{ Map_C(e,c^\prime)&\to&Map_C(e,d) \\ \downarrow&&\downarrow \\ Map_C(F(e),F(c^\prime))&\to&Map_C(F(e),F(d)) }$

is a homotopy pullback.

Proposition 2.4.1.13

Let $p:X\to S$ be an inner fibration of simplicial sets. Let $f:x\to y$ be an edge of $X$. Let $\sigma:\Delta^3\to X$ be a 3-simplex such that $d_1 \sigma= s_0 f$ and $d_2\sigma=s_1 f$. Let $\tilde f:\tilde x\to y$ be a $p$-cartesian edge such that $p(\tilde f)=p(f)$. Then $f$ is $p$-cartesian.

2.4.2 Cartesian fibrations

Definition 2.4.2.1

Let $p:X\to S$ be a map of simplicial sets. Then $p$ is called a cartesian fibration if the following coditions are satisfied.

1. $p$ is an inner fibration.

2. Every edge of has a $p$-cartesian lift.

Proposition 2.4.2.3
1. Any isomorphism of simplicial sets is a cartesian fibration.

2. The class of cartesian fibrations is closed under base change.

3. A composition of cartesian fibrations is a cartesian fibration.

2.4.7 Bifibrations

Corollary 2.4.7.12

Last revised on June 28, 2012 at 15:28:51. See the history of this page for a list of all contributions to it.